Symplectic Runge-Kutta methods satisfying effective … · Symplectic Runge-Kutta methods satisfying ... Therefore for symplectic Runge–Kutta method the second order condition is

Symplectic Runge-Kutta methods satisfying effective order conditions

Symplectic Runge-Kutta methods satisfyingeffective order conditions

Gulshad Imran

John Butcher (supervisor)

University of Auckland

11–15 July, 2011

International Conference on Scientific Computation andDifferential equations (SciCADE 2011)

University of Toronto, Canada

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Introduction

1 The idea of effective order was first introduced by Butcher in1969 for explicit Runge–Kutta methods as a mean ofovercoming the 5th order 5 stage barrier.

2 This was extended to Singly Implicit Runge–Kutta methodsby Butcher and Chartier in 1997.

3 The idea was later used for Diagonally Extended SinglyImplicit Runge–Kutta methods by Butcher and Diamantakisin 1998 and by Butcher and Chen in 2000 .

4 The accuracy of symplectic integrators for Hamiltoniansystems was enhanced using effective order by M ALopez-Marcos, J M Sanz-Serna and R D Skeel in 1996.

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Introduction

1 The idea of effective order was first introduced by Butcher in1969 for explicit Runge–Kutta methods as a mean ofovercoming the 5th order 5 stage barrier.

2 This was extended to Singly Implicit Runge–Kutta methodsby Butcher and Chartier in 1997.

3 The idea was later used for Diagonally Extended SinglyImplicit Runge–Kutta methods by Butcher and Diamantakisin 1998 and by Butcher and Chen in 2000 .

4 The accuracy of symplectic integrators for Hamiltoniansystems was enhanced using effective order by M ALopez-Marcos, J M Sanz-Serna and R D Skeel in 1996.

The purpose of this presentation is to develop some special

symplectic effective order methods with low implementation

costs.2/30


Differential Equations with Invariants

Consider an initial value problem

y ′(x) = f (y(x)), y(x0) = y0. (1)

Suppose Q(y) = yTMy is a quadratic invariant , that is

Q ′(y)f (y) = 0,

oryTMf (y) = 0.

Methods which conserve quadratic invariants are said to be

“Canonical” or “Symplectic”.

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Outline1 Effective order

Algebraic interpretation

Computational interpretation

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Computational interpretation2 Symplectic Runge–Kutta methods

Superfluous and Non–superfluous trees

Order conditions for Symplectic methods

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Order conditions for Symplectic methods3 Effective order with symplectic integrator

New Implicit Methods

Cheap implementation

Transformation

Starting method

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Transformation

Starting method4 Numerical Experiments

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Transformation

Starting method4 Numerical Experiments5 Conclusions

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Transformation

Starting method4 Numerical Experiments5 Conclusions6 Future work

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Effective order



1 We introduce an algebraic system which represents indvidualRunge–Kutta methods and also composition of methods.

2 This centres on the meaning of order for Runge–Kuttamethods and leads to the possible generalisation to the“effective order”.

3 We introduce a group G whose elements are mapping from T

(rooted –trees) to real numbers and where the groupoperation is defined according to the algebraic theory ofRunge–Kutta methods or to the theory of B–series.

4 Members of G represents Runge–Kutta methods with E

representing the exact solution. That is, E : T → R is definedby

E (t) = 1γ(t) , ∀t

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Effective order


Table: Group Operation

t r(t) α(t) β(t) (αβ)(t) E (t)

1 α1 β1 α1β0 + β1 1

2 α2 β2 α2β0 + β2 + α1β112

3 α3 β3 α3β0 + β3 + α21β1 + 2α1β2

13

3 α4 β4 α4β0 + β4 + α1β2 + α2β116

4 α5 β5 α5β0 + β5 + 3α1β3 + α21β1 + α3

1β114

4 α6 β6α6β0 + β6 + α1β4 + α1β3+ 1

8α21β2 + α2β2 + α1α2β1

4 α7 β7 α7β0 + β7 + 2α1β4 + α3β1 + α21β2

112

4 α8 β8 α8β0 + β8 + α2β2 + α1β4 + α4β1124

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We introduce Np as a normal subgroup, which is defined by

Np = {α ∈ G : α(t) = 0,whenever r(t) ≤ p}

A Runge- Kutta method with group element α is of order p, if it isin the same coset as ENp , that is

αNp = ENp

A Runge- Kutta method has an “effective order” p if there existanother Runge - Kutta method with corresponding group elementβ, such that

βαNp = EβNp

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The conjugacy concept in group theory provides a computationalinterpretation of the effective order. This means that, “every inputvalue for effective order method α is perturbed by a method β”.Therefore the starting method β offers some freedom of the orderconditions of the effective order method α.Every output value could also be perturbed back to the origionaltrajectory using method β−1.

x0 En

β

αn

xn

β−1

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Symplectic Runge–Kutta methods

Symplectic Runge– Kutta methods

A Runge–Kutta method is said to be canonical or symplectic if thenumerical solution yn also has the quadratic invariant Q(yn) i.e.

〈yn, yn〉 = 〈yn−1, yn−1〉

A method has this property if and only if,

biaij + bjaji − bibj = 0

for all i , j .

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It is a consequence of the symplectic condition that if τ1 and τ2 arerooted trees corressponding to the same tree τ then

φ(τ1) =1

γ(τ1), φ(τ2) =

1γ(τ2)

For Symplectic Runge–Kutta methods, we distinguish trees in twoways.

Superfluous trees,

Non– Superfluous trees.

τ11 2τ

τ210/30




Order conditions corresponding to non–superfluous trees aretransformed into one order condition.

a bτ

τa τb

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Order Conditions for Symplectic Runge–Kutta methods

The number of order conditions for symplectic Runge–Kuttamethods is less than the number of order conditions for a generalRunge–Kutta method.

Case 1: Suppose the method is of order at least 1, (∑

i

bi = 1),

∑

i ,j

biaij +∑

i ,j

bjaji −∑

i ,j

bibj = 0

⇒∑

i ,j

biaij =12

Therefore for symplectic Runge–Kutta method the second ordercondition is automatically satisfied and hence not required.

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Case 2 : Suppose the method is of order at least 2,

(∑

i

bici =1

2), Consider the symplectic condition,


Multiply with cj and take summation,

∑

i ,j

biaijcj +∑

i ,j

bjcjaji −∑

i ,j

bibjcj = 0

⇒(∑

i ,j

biaijcj −16) + (

∑

j

bjc2j − 1

3) = 0





Case 2 : Suppose the method is of order at least 2,

(∑

i

bici =1

2), Consider the symplectic condition,


Multiply with cj and take summation,

∑

i ,j

biaijcj +∑

i ,j

bjcjaji −∑

i ,j

bibjcj = 0

⇒(∑

i ,j

biaijcj −16) + (

∑

j

bjc2j − 1

3) = 0

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Order General RK method Symplectic RK method

1 1 1

2 2 1

3 4 2

4 8 3

Table: Order conditions for general and symplectic Runge–Kuttamethods up to order 4.

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Gauss method

For example, consider applying Gauss method to the harmonicoscillator problem given below:

q′ = p, p′ = −q.

The energy is given by,

H =p2

2+

q2

2.

The exact solution is,[

p(x)

q(x)

]

=

[

cos(x) −sin(x)

sin(x) cos(x)

][

p(0)

q(0)

]

.

This problem describes the motion of a unit mass attached to aspring with momentum p and position co-ordinates q defines aHamiltonian system.

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we consider the two stage order four Gauss method,

12 −

√36

14

14 −

√36

12 +

√36

14 +

√36

14

12

12

(2)

Gauss method approximatelyconserve the total energy ofthe himiltonian system .

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Effective order with symplectic integrator


We show a way of analyzing methods of effective order 4 havingsymplectic condition with three stages.

(βα)( ) = β( ) + α( )

(βα)( ) = β( ) + 2β( )α( ) + β2( )α( ) + α( )

(βα)( ) = α( ) + 3β( )α( ) + 3β2( )α( )

+ β3( )α( ) + β( )

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New Implicit methods

We present two examples of symplectic effective order methods:

Method A - has real and distinct eigenvalues

38

715 −163

50473315

58 −17

40 − 19

209180

1 1265

157234

1390

1415 −2

91345

Method B - has complex eigenvalues

16

314 − 1

21 034 −5

7 − 128 0

34 −3

7114

14

37

114

12

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Since method A have real eigenvalues, it is therefore ofinterest.This is because we can obtain a cheaper implementation.Here we are considering only method A.The general form of an s-stage implicit method is

yn+1 = yn + h

s∑

i=1

bi f (xn + hci ,Yi ),

Yi = yn + h

s∑

j=1

aij f (xn + hcj ,Yj)

The stage equations can be written in the form

Y = e ⊗ yn + h(A⊗ Im)F (Y )

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We use modified Newton Raphson iteration scheme to solve theabove equation. This can be defined as

M∆Y [k] = G (Y [k])

Y [k+1] = Y [k] +∆Y [k]

where

M = Is ⊗ Im − h(A⊗ J)

G (Y [k]) = −Y [k] + e ⊗ yn + h(A⊗ Im)F (Y )

The total computational cost in this scheme include

the evaluation of F and G ,

the evaluation of J,

the evaluation of M,

LU factorization of the iteration matrix, M,

back substitution to get the Newton update vector, ∆Y [k].

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Transformation

Transformation

To reduce computational cost of fully implicit RK method, we usetransformation. The transformation matrix T for method A isgiven by

T =

0.262527404618574 0.949059020237884 −0.235024483067430

0.812033424383500 −0.068304645470165 0.765241433855123

−0.521230351675958 0.307606000449118 0.599307133505220

The transformation matrix has the property,

T−1AT =

−0.993809382166128 0 0

0 0.565055763297954 0

0 0 0.928753618868175

where −0.993809382166128, 0.565055763297954, and0.928753618868175 are three distinct real eigenvalues

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Starting method

Starting method

Solution of these equations give the starting method

(α)( ) = 1

(α)( ) = 2β( ) + 13

(α)( ) = 3β( ) + 3β( ) + 14

which is given by

2 134608

45952304

134608

0 46214608 − 13

2304 −45954608

−2 134608 − 4621

230413

4608

132304 − 13

115213

2304

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Numerical Experiments


1 The Kepler’s problem

x ′1 = y1, x ′2 = y2,

y ′1 = −x1

(x21 + x22 )32

, y ′2 = −x2

(x21 + x22 )32

where (x1, x2) are the position coordinates and (y1, y2) are thevelocity components of the body.

(x1, x2, y1, y2) = (1− e, 0, 0,√

(1 + e)/(1− e)),

H = 12(y

21 + y22 )−

1√

x21 + x22

.

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Kepler problem (e=0,h=0.01, n = 106)

Graph for Hamiltonian:error VS time

0 1000 2000 3000 4000 5000 6000−5

0

5

10

15

20x 10

−14

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Kepler problem (e=0.5,h=0.01, n = 106)

For Hamiltonian:error VS time

0 1000 2000 3000 4000 5000 6000−5

0

5

10

15

20x 10

−10

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1 The simple Pendulum

p′ = − sin(q), q′ = p,

(p, q) = (0, 2.3).

H =p2

2− cos(q).

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Simple Pendulum (h=0.05, n = 106)

For Hamiltonian:error VS time

0 1 2 3 4 5

x 104

−1

0

1

2

3

4

5

6x 10

−8

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Conclusions

Conclusions

1 For problems that conserve some sort of invariant structure, itis a good idea to use numerical methods which mimic thisbehaviour.

2 Symplectic Runge–Kutta methods have this role for manyimportant problems.

3 Because of greater flexibilty,effective order methods canprovide greater efficiency as compared with methods withclassical order.

4 It is possible to obtain cheap implementation cost if A hasreal eigenvalues.

5 These methods are suited for parallel computers which havevery large number of processors.

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Future work

Future work

1 Error estimates

2 Working on implicit methods with optimal choices ofparameters.

3 Construct general linear methods with closely relatedproperties.

4 Generalization of effective order on partioned Runge–Kuttamethods for separable Hamiltonian.

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Future work

THANK YOU

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Documents

Symplectic Runge-Kutta methods satisfying effective … · Symplectic Runge-Kutta methods satisfying ... Therefore for symplectic Runge–Kutta method the second order condition is